A Simple and Easy Math Problem

On June 13th we returned to the Cape Akrotiri lighthouse where we first looked at the domes that formed the Akrotiri peninsula. This time, Lisa wanted to show us the most massive block she’s ever seen. A block is a solidified rock that is thrown up into the air during an eruption, and can be any shape. This blocks shape was gigantic.

We walked down the slope behind the Akrotiri lighthouse. Lisa pointed out the thin layer of pumice from Phase 1 of the eruption which consisted of pumice fall. On top of Phase 1 was the white lapilli and ash beds of Phase 2 that were deposited as a result of pyroclastic surges. Pyroclastic surges are turbulent clouds of ash and lapilli and lithic fragments that tumble across the landscape outward from the vent depositing the material according to density. Lithic fragments are usually deposited together in a pyroclastic surge because of their density. The giant block Lisa brought us down there to see was obviously too large to be carried by a pyroclastic surge. It wasn’t even close to the size of the lithic fragments carried by the surge. So how did it get there?

The only way for the giant block to get there would for it to be launched from the vent area in an explosion. Throughout the phases of the Minoan eruption the vent was excavated, growing larger and larger with time. The explosions blew up the vent and sent pieces of rock flying up in to the air, like the giant block. This block was abnormally large. Lisa asked us all to examine its surface and determine its composition, and think about how it got there. I decided to measure it. Its width was three meters wide by two meters tall by two meters deep: a large rock. From the GIS tool on the iPads we determined the block was thrown 6.3 kilometers (4.05 miles) from the vent area at Nea Kameni: a large explosion.

Lisa with the block.

Lisa offhandedly challenged the group with a question which was “How much force was require to throw this thing?” Thus beginning a frenzy of geologists attempting physics. Sheridan, Sarah, Leah, Lisa and I all got to work discussing what we needed in order to do this calculation. We finished each others sentences and thoughts as our brains scrambled to figure it out. “Force is mass times acceleration…” I said. Then Lisaadded, “and mass is density and volume,” Sheridan added, “No mass is volume times density.” “Is it?” we’d all say. We all had no idea what we were really talking about. But, we had the length, width, and height. The group determined the rock was basalt, and thank god some person before us had determined the density of basalt which Lisa said was 2.9 g/cm^3. We had a starting point, so I got to work calculating the mass.

We came to a stand still in determining the other component Newton’s second law: acceleration. Sarah was trying to tell us that objects in projectile motion only have acceleration working on them in the y direction, gravity. But none of us believed her because we’re geologists not physicists. We looked up the kinematic equations trying to find the best way to calculate acceleration. There were too many variables that we couldn’t determine. We needed to know the time, the starting velocity, the maximum height of this projectile arc, too many things that we would have had to speculate. Sheridan, Lisa and I tried to calculate the angle at which is was thrown. Knowing the angle would help us determine the starting velocity. Objects that are projected into the air have velocity in the x-direciton and the y-direction and using the inverse tangent of this ratio you can find the actual starting velocity. We determined the horizontal distance the block was thrown by using GIS on the iPads. The horizontal distance the block was thrown was 6.3 kilometers or 6,300 meters. If we could just find the elevation of Cape Akrotiri (the height or y-direction displacement) we could determine the angle.

The physics frenzy came to a halt when we ran into the obstacle of finding the elevation at Cape Akrotiri. There was a lot of information missing in this problem, a lot of things I’d need to look up and find out later on.

Some goofy geology student standing on giant block

One way a geologist would find the force, distance, flight time, or any information about this block would be to measure the amount of block sag. Block sag is the depth of the impact recorded in the rock record. When blocks hit the ground in an eruption they usually don’t bounce right back up; they sink into the the ground due to the weakness or in this case wetness of the unconsolidated ground. Since the Minoan eruption was greatly impacted by water, most of the deposits were wet when they were deposited due to the high water content in the magma. When this block fell it sagged all the way through Phase 1 and Phase 0 down to the Cape Riva Tuff from Santorini’s third caldera forming eruption 21,000 years ago. It sagged through rock and time. However, the block sag was impossible to measure due to erosion and the topography in the area where it fell, thus math was the only way.

…

When I sat down to work on this problem the other day I finally realized what Sarah was saying about acceleration. The block was launched from the vent and flew through the air. No other force acted on it after the launch, nothing was accelerating the block and speeding it up. Nothing was slowing it down either except for the force of gravity pulling it back down to earth’s surface. So, the only acceleration the block had was acceleration due to gravity. This made solving for the force on the block much easier.

For the sake of this problem and the limited surface area of the block exposed, Sheridan suggested treating the block as a cylinder to find the volume. But upon further thought we decided comparing the block to a sphere would yield more accurate results.

Force is equal to mass multiplied by acceleration, and mass is density times volume. Using the volume for a sphere equation and cutting the 3 meter width in half to find the “radius” we can find the volume. The density can be found using the given density of basalt which is 2.9 g/cm^3. To put this into SI units we multiply by 10^3 to get kg/m^3. Multiplying these two found variables together we get 40,997.3 kg for the mass of the block.

To find force we multiply the mass, 40,997.3 kg by 9.8 m/s^2. This gives us 401,773.54 kg*m/s^2 or another name for this unit of measure is newtons. The force required to throw this block was 401,773.54 newtons. For comparison the thrust force required to lift Apollo 11 off the ground was 33,400,000 newtons (1). The thrust force of Apollo 11 was only 83 times greater than the force required to throw this block. For further comparison the force of a 3,200 lb car crashing into a tree at 30 mph results in a force of 428,514 newtons (2), which is roughly the same amount of force as the force on the block.

Next, I wanted to determine the velocity at which it was thrown. This is somewhat tedious because in order to find initial velocity I’d need the angle at which it was thrown. At first I thought it was possible because I had the horizontal distance and vertical height. However solving for inverse tangent gave me an angle of .01 degrees which just seemed improbable. So I chose to make a guess at it and solve for velocity with three hypothetical angles, yielding three hypothetical scenarios.

This was the only equation I could use with the given information I had. I solved for velocity by using the range found on the iPads (6300 m), and then worked backward from there with the other three equations.

Now that I had a velocity for Scenarios A, B and C, I could use that to find the maximum height. This equation solves for the maximum height an object would reach in its projectile arc. After it reaches its top height, (provided by its initial velocity), the object falls back down to earth. Gravity never stops pulling it back down, its constantly forcing it to fall to earth.

Next I decided to solve for time. This block was thrown 6.3 kilometers in the air at three possible velocities with three possible heights, but the most shocking thing to me was how much time it took to get there. For example with scenario C? 28.89 seconds?! What the schist honestly. Time is dependent on the angle and speed at which it is thrown. Scenario C has the greatest angle (33 degrees), so it was launched further up in the air than in the other scenarios, it would have to take more time.

Here is a picture of my notes on Scenarios A, B, and C. For Scenario A I used an angle of eight degrees, because it is possible the block was shot out laterally from the vent. An angle of eight degrees yields a velocity of 473.31 m/s, the fastest velocity out of the three scenarios. It would reach a height of 221.1 meters above sea level and it would take a total of 13.44 seconds to get to Cape Akrotiri. Although all this is possible, it doesn’t seem probable.

Scenario B is a more likely scenario. For Scenario B I chose an angle of 16 degrees. At this angle the block would have reached a height of 451.62 meters, and would have arrived after 19.20 seconds. Scenario B seems closer to what actually occurred. A steeper angle results in a greater height but a smaller velocity. Based on the small, (not remotely accurate) graph I drew Scenario B seems like it could have actually happened.

Scenario C is based on an angle of 33 degrees. With this angle the block would have moved at 260 m/s, reached a height of 1023 meters, and taken 30 seconds to arrive. This seems like an incredible feat, it shows the immensity of this eruption. The force required to launch something this big six kilometers in the air, strikes me with awe.

The Minoan Eruption occurred in less than 24 hours. It’s the second largest eruption in history following Yellowstone. The Minoan Eruption deposited greater than 30 meters of pumice, ash, and pyroclastic material that built up the sides of the caldera rim to the elevation it is today. This eruption had an eruption column that reached 50 kilometers high into the stratosphere, it launched 14 cubic meter blocks of basalt 6 kilometers in the air. This eruption completely destroyed and rebuilt the landscape of this island. Someday, all the evidence of this eruption, of this entire island, will be preserved in the rock record as meter bedded layers of pyroclastic material. The caldera rim will be preserved in rock, the entire topography of this island will someday be buried. Maybe some geologist in the future, if humans still exist at that time, will look at the rock layers of Santorini and be impressed, but maybe not. We are lucky that we exist at this time to see the evidence of such a cataclysmic event still fresh on the landscape.

Nature is the most important thing in our world, its the only thing in our world. Nothing else matters; not money, not countries, not even people. Nature dominates and overcomes it all. I love geology because it puts our tiny lives in perspective, and it’s comforting to know that nothing in our lives has any significance at all. Our world, our global civilization will one day be a couple meters of plastic and garbage and metal in the rock record. The dinosaurs dominated earth for 170 million years and we’ve only been here for three. Look at what we’ve done to Earth in such a short time. The dinosaurs, though they were giants who roamed this earth, still succumbed to nature and went extinct.

I agree with the comment above — you explained a very complicated (and intimidating!) problem in a way that I (a nondisciplinary audience) could understand. Explaining the phases you discuss in the beginning would have been helpful for me. I enjoyed reading your post; your narrative style made your blog engaging. Bravo!

Emily – I didn’t think it was possible for you to write this blog without completely boring your reader, but you did a really great job. Excellent work. I feel like the mass of the block is overestimated….I mean that is equivalent to 45 tons.

The Minoan Eruption was the second largest in human history following Tambora – please correct that.

While your conclusion is definitely heart felt, I thought it was a little out of place in the context of this blog. The tone and style changed in that paragraph and it would have benefited from a wrap up/summary of the struggle to solve this problem